[SOLVED] Characteristic Function - Interior and Exterior point

Problem:
Let A be a subset of . The function defined by

is called the characteristic function of A. Show that exists if and only if x* is either an interior or exterior point of the set A.
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Since A is a subset of , then let x* be a point in .

If x* is an interior point then there is an open ball about x* that is contained in A.

My question is that if x* is an interior point, then the since x* is contained in A therefore ?

Similarly, if x* is an exterior point(provided that there is an open ball about x* that is contained in , then since